The Hölder mean with exponential $p$ is defined as

or a weighted version with $w=w_1+w_2+\cdots+w_n$

which for $p=\infty$ and $p=-\infty$, then

Hölder mean is a generalization of means:

  • $p=-\infty$: minimum
  • $p=-1$: Harmonic mean
  • $p=0$: Geometric mean (obtained by taking limit on $p\to 0$)
  • $p=1$: Arithmetic mean
  • $p=2$: Root-mean-square
  • $p=\infty$: maximum

The Hölder mean has the property that, $M_p \le M_q$ for $p<q$ and the equality holds only when $x_1=x_2=\cdots=x_n$, or, for the weighted version, $w_1=w_2=\cdots=w_n$ and $x_1=x_2=\cdots=x_n$. This is the generalization of the mean inequality of $\textrm{A.M.} \ge \textrm{G.M.} \ge \textrm{H.M.}$